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21: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. These asymptotic expansions are in fact convergent expansions. For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …

22: Bibliography S

… ►

Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

Encodings:

BibTeX

… ►

Error bounds for a uniform asymptotic expansion of the Legendre function $P_n^{-m}(\cosh z)$ . Quart. Appl. Math. 46 (3), pp. 473–488.

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External Links:

ISSN 0033-569X, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

…

23: 11.10 Anger–Weber Functions

… ►

11.10.19 $${𝐉}_{-\frac{1}{2}}\left(z\right)={𝐄}_{\frac{1}{2}}\left(z\right)={(\frac{1}{2}\pi z)}^{-\frac{1}{2}}(A_{+}(\chi )\cos z-A_{-}(\chi )\sin z),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $A$: expansion function and $\chi$

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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11.10.20 $${𝐉}_{\frac{1}{2}}\left(z\right)=-{𝐄}_{-\frac{1}{2}}\left(z\right)={(\frac{1}{2}\pi z)}^{-\frac{1}{2}}(A_{+}(\chi )\sin z+A_{-}(\chi )\cos z),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $A$: expansion function and $\chi$

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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24: 7.13 Zeros

… ►As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $C\left(z\right)$ satisfy … ►For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971). …

25: 20.2 Definitions and Periodic Properties

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20.2.1 $${\theta }_1\left(z\right|\tau )={\theta }_1(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n{q}^{{(n+\frac{1}{2})}^2}\sin \left((2n+1)z\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\sin z$: sine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.1

Referenced by:

§20.14, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

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20.2.2 $${\theta }_2\left(z\right|\tau )={\theta }_2(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{q}^{{(n+\frac{1}{2})}^2}\cos \left((2n+1)z\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.2

Permalink:

http://dlmf.nist.gov/20.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

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20.2.3 $${\theta }_3\left(z\right|\tau )={\theta }_3(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\cos \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.3

Referenced by:

§20.10(i), §20.13, §20.14, §27.13(iv)

Permalink:

http://dlmf.nist.gov/20.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. …

26: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►

28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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27: 10.67 Asymptotic Expansions for Large Argument

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10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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10.67.3 $${\mathrm{ber}}_{\nu }x\sim \frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi \right)-\frac{1}{\pi }(\sin \left(2\nu \pi \right){\ker }_{\nu }x+\cos \left(2\nu \pi \right){\mathrm{kei}}_{\nu }x),$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.1, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(i), §10.67(i), §10.68(iii), §10.69

Permalink:

http://dlmf.nist.gov/10.67.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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10.67.4 $${\mathrm{bei}}_{\nu }x\sim \frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\sin \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi \right)+\frac{1}{\pi }(\cos \left(2\nu \pi \right){\ker }_{\nu }x-\sin \left(2\nu \pi \right){\mathrm{kei}}_{\nu }x).$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.2, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(i), §10.67(i), §10.67(ii), §10.68(iii), §10.69

Permalink:

http://dlmf.nist.gov/10.67.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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10.67.5 $${\ker }_{\nu }^{\prime }x\sim -{\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}-\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Permalink:

http://dlmf.nist.gov/10.67.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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10.67.7 $${\mathrm{ber}}_{\nu }^{\prime }x\sim \frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{3k}{4}+\frac{1}{8}\right)\pi \right)-\frac{1}{\pi }(\sin \left(2\nu \pi \right){\ker }_{\nu }^{\prime }x+\cos \left(2\nu \pi \right){\mathrm{kei}}_{\nu }^{\prime }x),$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $b_k(\nu )$: polynomial coefficient

A&S Ref:

§9.10 (modified)

Referenced by:

§10.67(i), §10.67(i), §10.69

Permalink:

http://dlmf.nist.gov/10.67.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

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28: 28.6 Expansions for Small $q$

… ►For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ everywhere in (28.6.26). …

29: 6.18 Methods of Computation

… ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. … ►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …

30: 18.15 Asymptotic Approximations

… ►Asymptotic expansions for $C_n^{(\lambda )}\left(\cos \theta \right)$ can be obtained from the results given in §18.15(i) by setting $\alpha =\beta =\lambda -\frac{1}{2}$ and referring to (18.7.1). … ►For asymptotic expansions of $P_n\left(\cos \theta \right)$ and $P_n\left(\cosh \xi \right)$ that are uniformly valid when $0\le \theta \le \pi -\delta$ and see §14.15(iii) with $\mu =0$ and $\nu =n$. …